/export/starexec/sandbox2/solver/bin/starexec_run_complexity /export/starexec/sandbox2/benchmark/theBenchmark.xml /export/starexec/sandbox2/output/output_files


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WORST_CASE(Omega(n^1), O(n^1))
proof of /export/starexec/sandbox2/benchmark/theBenchmark.xml
# AProVE Commit ID: 48fb2092695e11cc9f56e44b17a92a5f88ffb256 marcel 20180622 unpublished dirty


The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).

(0) CpxTRS
(1) NestedDefinedSymbolProof [UPPER BOUND(ID), 0 ms]
(2) CpxTRS
    (3) RelTrsToTrsProof [UPPER BOUND(ID), 0 ms]
    (4) CpxTRS
    (5) CpxTrsMatchBoundsTAProof [FINISHED, 85 ms]
    (6) BOUNDS(1, n^1)
(7) RelTrsToDecreasingLoopProblemProof [LOWER BOUND(ID), 0 ms]
(8) TRS for Loop Detection
    (9) DecreasingLoopProof [LOWER BOUND(ID), 0 ms]
    (10) BEST
        (11) proven lower bound
            (12) LowerBoundPropagationProof [FINISHED, 0 ms]
            (13) BOUNDS(n^1, INF)
        (14) TRS for Loop Detection


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(0)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL
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(1) NestedDefinedSymbolProof (UPPER BOUND(ID))
The following defined symbols can occur below the 0th argument of top: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of active: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of f: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of start: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of match: active, start, proper, top, f, check, match
The following defined symbols can occur below the 0th argument of check: active, start, proper, top, f, check, match

Hence, the left-hand sides of the following rules are not basic-reachable and can be removed:
match(f(x), f(y)) -> f(match(x, y))
proper(f(x)) -> f(proper(x))

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(2)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL
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(3) RelTrsToTrsProof (UPPER BOUND(ID))
transformed relative TRS to TRS
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(4)
Obligation:
The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL
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(5) CpxTrsMatchBoundsTAProof (FINISHED)
A linear upper bound on the runtime complexity of the TRS R could be shown with a Match-Bound[TAB_LEFTLINEAR,TAB_NONLEFTLINEAR] (for contructor-based start-terms) of 3. 

The compatible tree automaton used to show the Match-Boundedness (for constructor-based start-terms) is represented by: 
final states : [1, 2, 3, 4, 5, 6, 7]
transitions: 
mark0(0) -> 0
c0() -> 0
X0() -> 0
ok0(0) -> 0
found0(0) -> 0
active0(0) -> 1
top0(0) -> 2
check0(0) -> 3
match0(0, 0) -> 4
proper0(0) -> 5
f0(0) -> 6
start0(0) -> 7
check1(0) -> 8
top1(8) -> 2
X1() -> 11
f1(11) -> 10
match1(10, 0) -> 9
start1(9) -> 3
proper1(0) -> 4
c1() -> 12
ok1(12) -> 5
f1(0) -> 13
ok1(13) -> 6
found1(0) -> 7
f1(0) -> 14
found1(14) -> 6
active1(0) -> 15
top1(15) -> 2
f1(0) -> 16
mark1(16) -> 6
c1() -> 18
mark1(18) -> 17
top1(17) -> 2
X2() -> 21
f2(21) -> 20
match2(20, 0) -> 19
start2(19) -> 8
ok1(12) -> 4
ok1(13) -> 13
ok1(13) -> 14
ok1(13) -> 16
found1(14) -> 13
found1(14) -> 14
found1(14) -> 16
mark1(16) -> 13
mark1(16) -> 14
mark1(16) -> 16
check2(18) -> 22
top2(22) -> 2
X3() -> 25
f3(25) -> 24
match3(24, 18) -> 23
start3(23) -> 22

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(6)
BOUNDS(1, n^1)

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(7) RelTrsToDecreasingLoopProblemProof (LOWER BOUND(ID))
Transformed a relative TRS into a decreasing-loop problem.
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(8)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL
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(9) DecreasingLoopProof (LOWER BOUND(ID))
The following loop(s) give(s) rise to the lower bound Omega(n^1):

The rewrite sequence

f(found(x)) ->^+ found(f(x))

gives rise to a decreasing loop by considering the right hand sides subterm at position [0].

The pumping substitution is [x / found(x)].

The result substitution is [ ].




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(10)
Complex Obligation (BEST)

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(11)
Obligation:
Proved the lower bound n^1 for the following obligation:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL
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(12) LowerBoundPropagationProof (FINISHED)
Propagated lower bound.
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(13)
BOUNDS(n^1, INF)

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(14)
Obligation:
Analyzing the following TRS for decreasing loops:

The Runtime Complexity (full) of the given CpxTRS could be proven to be BOUNDS(n^1, n^1).


The TRS R consists of the following rules:

   active(f(x)) -> mark(x)
   top(active(c)) -> top(mark(c))
   top(mark(x)) -> top(check(x))
   check(f(x)) -> f(check(x))
   check(x) -> start(match(f(X), x))
   match(f(x), f(y)) -> f(match(x, y))
   match(X, x) -> proper(x)
   proper(c) -> ok(c)
   proper(f(x)) -> f(proper(x))
   f(ok(x)) -> ok(f(x))
   start(ok(x)) -> found(x)
   f(found(x)) -> found(f(x))
   top(found(x)) -> top(active(x))
   active(f(x)) -> f(active(x))
   f(mark(x)) -> mark(f(x))

S is empty.
Rewrite Strategy: FULL